Mathematical Sciences: Weighted Norm Inequalities and Partial Differential Equations

该项目涉及两个分领域的问题， 数学分析：偏微分方程和 与高斯测度相关的调和分析。 项目 有两个目标。 第一个是分析一类解的行为 一类散度型退化抛物方程 非平滑系数 这样的系数可以是零、无穷大 或两者 已知该方程用于模拟 在非均匀和非各向同性材料中的温度。 在这项工作中要考虑的基本问题包括： 暗示解的正则性的条件，确定 Harnack原理是否对非负解有效 并理解基本解的行为 方程式 Poincare不等式和 本研究将使用Sobolev型。 第二个目标涉及的行为，在空间， 一类关于高斯测度可积的函数 由Ornstein-Uhlenbeck半群生成的变换。 在高斯背景下，这些变换扮演着 M.的经典奇异积分里斯 特别感兴趣的 将是建立弱型估计与常数 独立于维数有界。 这次调查将 集中在使用最大函数，如在卡尔德龙- Zygmund理论，但这个函数并不继承所有的 经典极大函数的性质。 不过有 有证据表明，它仍然是弱型， 高斯测度 这是第一个要解决的问题。

This project is concerned with problems in two subareas of mathematical analysis: partial differential equations and harmonic analysis related to the Gaussian measure. The project has two objectives. The first is to analyze the behavior of solutions of a class of degenerate parabolic equations in divergence form with non-smooth coefficients. Such coefficients may be zero, infinite or both. The equations are known to model cases of diffusion of temperature in a non-homogeneous and non-isotropic material. Basic questions to be considered in this work include finding conditions which imply regularity of solutions, determining whether a Harnack principle is valid for non-negative solutions and understanding the behavior of the fundamental solutions of the equations. Weighted norm inequalities of the Poincare and Sobolev type will be used in this study. The second objective concerns the behavior, in the space of functions integrable with respect to Gaussian measure, of a class of transforms generated by the Ornstein-Uhlenbeck semigroup. These transforms play the role, in the Gaussian context, of the classical singular integrals of M. Riesz. Of particular interest will be the establishment of weak-type estimates with constants bounded independently of dimension. This investigation will center on the use of a maximal function as in the Calderon- Zygmund theory, but this function does not inherit all the properties of the classical maximal functions. However, there is evidence to suggest that it is still of weak type with respect to Gaussian measure. This is the first problem to be resolved.